Stability of simple periodic orbits and chaos in a Fermi-Pasta-Ulam lattice

We investigate the connection between local and global dynamics in the Fermi-Pasta-Ulam (FPU) beta model from the point of view of stability of its simplest periodic orbits (SPO's). In particular, we show that there is a relatively high-q mode [q = 2(N + 1)/3] of the linear lattice, having one particle fixed every two oppositely moving ones (called SPO2 here), which can be exactly continued to the nonlinear case for N = 5 + 3m, m = 0,1,2,, and whose first destabilization E(2u), as the energy (or beta) increases for any fixed N, practically coincides with the onset of a "weak" form of chaos preceding the breakdown of FPU recurrences, as predicted recently in a similar study of the continuation of a very low (q = 3) mode of the corresponding linear chain. This energy threshold per particle behaves like E(2u)/N alpha N(-2). We also follow exactly the properties of another SPO [with q = (N + 1)/2] in which fixed and moving particles are interchanged (called SPO1 here) and which destabilizes at higher energies than SPO2, since E(1u)/N alpha N(-1). We find that, immediately after their first destabilization, these SPO's have different (positive) Lyapunov spectra in their vicinity. However, as the energy increases further (at fixed N), these spectra converge to the same exponentially decreasing function, thus providing strong evidence that the chaotic regions around SPO1 and SPO2 have "merged" and large-scale chaos has spread throughout the lattice. Since these results hold for N arbitrarily large, they suggest a direct approach by which one can use local stability analysis of SPO's to estimate the energy threshold at which a transition to ergodicity occurs and thermodynamic properties such as Kolmogorov-Sinai entropies per particle can be computed for similar one-dimensional lattices.